Math notes/Pre-Calculus/Functions Review

Writing an Absolute Value Piecewise — Cheat sheet

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Key method

|u|=\begin{cases}u,&u\ge0,\\-u,&u<0.\end{cases}

For an expression $u(x)$ , solve $u(x)=0$ to locate the breakpoints before splitting into intervals.

Example

For $|2x-6|$ , the breakpoint is $x=3$ . If $x<3$ , then $2x-6<0$ and the value is $6-2x$ . If $x\ge3$ , the value is $2x-6$ . Both branches meet at zero when $x=3$ , producing a corner rather than a discontinuity.

Avoid this mistake

$|a+b|$ is not generally $|a|+|b|$ . For $a=1,b=-1$ , the two sides are $0$ and $2$ .